A Conservative Finite Element Solver for MHD Kinematics equations: Vector Potential method and Constraint Preconditioning

A new conservative finite element solver for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations is presented.The solver utilizes magnetic vector potential and current density as solution variables, which are discretized by H(curl)-conforming edge-element and H(div)-conforming face element respectively. As a result, the divergence-free constraints of discrete current density and magnetic induction are both satisfied. Moreover the solutions also preserve the total magnetic helicity. The generated linear algebraic equation is a typical dual saddle-point problem that is ill-conditioned and indefinite. To efficiently solve it, we develop a block preconditioner based on constraint preconditioning framework and devise a preconditioned FGMRES solver. Numerical experiments verify the conservative properties, the convergence rate of the discrete solutions and the robustness of the preconditioner.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1712.0892

Auxiliary Space Preconditioners for Mixed Finite Element Methods

Summary. This paper is devoted to study of an auxiliary spaces preconditioner for H(div) systems and its application in the mixed formulation of second order elliptic equations. Extensive numerical results show the efficiency and robustness of the algorithms, even in the presence of large coefficient variations. For the mixed formulation of elliptic equations, we use the augmented Lagrange technique to convert the solution of the saddle point problem into the solution of a nearly singular H(div) system. Numerical experiments also justify the robustness and efficiency of this scheme

Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media

In this paper we propose and analyze a preconditioner for a system arising from a finite element approximation of second order elliptic problems describing processes in highly het- erogeneous media. Our approach uses the technique of multilevel methods and the recently proposed preconditioner based on additive Schur complement approximation by J. Kraus (see [8]). The main results are the design and a theoretical and numerical justification of an iterative method for such problems that is robust with respect to the contrast of the media, defined as the ratio between the maximum and minimum values of the coefficient (related to the permeability/conductivity).Comment: 28 page

Mixed finite element and stochastic Galerkin methods for groundwater flow modelling: efficiency analysis and real-life application

The thesis concludes with the development of a numerical model for a real case study in the United Kingdom, which is one of the first examples of formal characterization of model uncertainty for an actual site